Error correcting codes are used in data recording systems to ensure data reliability. Parity codes are examples of error correction codes. Parity codes are often used to correct randomly occurring errors.
Short parity codes typically provide good error correction performance, but at a low code rate. Longer parity codes provide a higher code rate. However, high rate parity codes typically have reduced error correction performance and are more likely to propagate errors.
These deficiencies can be overcome by using a tensor product parity code or codes modified from a tensor product code. A tensor product parity (TPP) code is the tensor product of two smaller codes. The parity check matrix of a TPP code is derived by taking the tensor product of the parity check matrices for the two smaller codes.
For example, a tensor product parity code can be the tensor product of a short parity code and a component Bose-Chaudhuri-Hochquenghem (BCH) code. Such a tensor product parity code has an error correction performance equivalent to a short parity code, but with a substantially higher code rate. BCH codes are another well-known family of error correcting codes. The component BCH code can be replaced by any other error correction codes.
A Reed-Solomon (RS) error correction code can be combined with a tensor product parity (TPP) code to generate a combined code. The combined code can be used to provide two levels of error correction in a data recording system. While efficient encoding methods exist for encoding the TPP and RS codes separately, no such efficient encoder exists to simultaneously enforce both TPP and RS parity rules. The only method known to encode a combination RS and TPP code is by brute-force matrix multiply. Combined RS/TPP codes typically have very large parity check matrices, and as a result, they are difficult to encode, because they require extensive matrix multiplication.
It would therefore be desirable to provide combined error correcting codes that are simpler to encode and that require less extensive matrix multiplication.